The complete analysis of the curve-shortening flow for embedded closed curves, due to Gage-Hamilton [1]} and
Grayson [2]}, was the one of earliest successes in the theory of geometric flows. However, some years
before, motivated in part by the desire to understand dynamics of grain boundaries, Brakke [3]} defined a weak
form of this curvature flow in the class of varifolds, which thus provides one avenue for studying the evolution of
certain singular geometric objects. This weak formulation yields global existence of solutions, but on the other hand, does
not allow one to control the geometry of these evolving objects.
This has motivated much further study. We mention the work of Kim and Tonegawa [4]}, who showed
that one can obtain a nontrivial mean curvature flow even when the initial datum is singular, cf. also the recent
monograph [5]}. However, it is worth emphasizing that questions concerning uniqueness or non-uniqueness
of solutions are mostly inaccessible in this measure theoretic context.
